This blog post is my review of Godel Escher Bach: An Eternal Golden Braid by Douglas Hofstadter, although to be technical this is really more of a summary/ attempt to convince you to read this book. The only real review I have of this book is that it is brilliant and everyone should read it.

I picked up this book a year and a half ago thanks to Steve Yegge's blog post, and this is one of the smartest books I have ever read. When I say smart, I don't mean the subject material of the book is difficult (although it sure is), or reading the book makes you smart (although it probably will help). I mean that the author and the book, not the reader, are smart in the structure and content of the dialogues, the organization of the book, the braid he weaves connecting all these different topics, and just the various ways the book manages to refer to itself or talk about itself.

My first read through of this book was during my lunch breaks at OTC Markets, and after reading this book I started to read some reviews of it. One of the reviews that I read linked me to a retrospective by Hofstadter, who wrote 20 years after the book was published that most people misunderstood his book as a random collection of things he finds interesting, and about how "math, art, and music are totally the same thing at the core man!!!" I found this tremendously funny and a little self gratifying since I thought I was clearly not one of those people. Months later, when I was recommending this book to a friend, he asked me what the book was about.

Confidently, I answered- "Oh, this book is about how Godel, Escher, and Bach's ideas are connected and interwoven and how systems are limited and there's some stuff about Godel's incompleteness theorem and minds and each chapter is preceded by a dialogue between Achilles and a tortoise..."

And that was when I realized I needed to reread the book.

So this is my second attempt at reading through GEB, with the goal of really getting what this book is about. This time, while I was reading, every time I'd finish reading a chapter or two I would take notes on what I'd learned from the chapter (they are messy, but they are unedited here). I planned on writing this post from the very start of my read through, and many times I felt like I should give up. One of the reasons is because Hofstadter is much smarter than I am, and I have none of his wit, intelligence, or way with words, so what he says in a super dense book I am surely incapable of capturing in a short blog post. I also felt like I was doing the book a big injustice, and I wanted to avoid condensing the book in a crappy lossy way, losing a lot of its beauty and brilliance in the meanwhile.

I decided to keep going partly because I was already halfway, but mostly because I felt like it was helping me understand the book better. Perhaps that is why this blog post is a little messy still- I just couldn't make up my mind what I wanted this post to be or who I was writing this post for. But my original goal was threefold:

to better understand GEB

to convince you to give the book a shot

to be corrected if I get anything wrong

So what is GEB about?GEB is, as Hofstadter describes it, a "very personal attempt to say how it is that animate beings can come out of inanimate matter." It is an attempt to describe, postulate, and perhaps understand how consciousness and intelligence and notions of beauty and truth emerge from inflexible, hardwired neurons. How does the concept of "self" emerge from things as selfless as stones or ants? This question is key to AI, because before we attempt to program "artificial" intelligence, we must first understand what "real" intelligence is, and where it comes from.

I hope to roadmap parts of his attempt here.

The first half of the book, dubbed GEB, lays down some of the groundwork of the book, and prepares us for understanding Godel's Incompleteness Theorems.

Any formal system able to do elementary arithmetic is either inconsistent or incomplete.

Any formal system able to express its own consistency can prove its own consistency if and only if it is inconsistent.

This is explained in detail in the book, I am both unwilling and unable to explain it here.

Some precursors include the notion of formal systems, with a natural subgoal being establishing a formal system that can express all statements of number theory with rules that correspond to our reasoning. The idea of theorems & non theorems is linked to the idea of figure and ground in art (especially in Escher's work). He also looks at notions and origins of meaning, introducing propositional calculus and its extension, a formal system he calls Typographical Number Theory (TNT). TNT is created to be capable of expressing all number theoretic reasoning in a formal system via rigid symbol manipulation.

The next step is to "destroy" TNT as part of his proof of Godel's Incompleteness Theorem.

To do so, Hofstadter introduces the idea of "Godel numbering" of a system, of mapping symbols of a formal system onto numbers. This is a key part of the proof allowing us to recreate the Liar's Paradox ("This sentence is false", more detail here) in TNT by allowing strings of TNT to talk about TNT. More formally, we can make statements about formal systems expressible in numbers, so expressible in number theory, so expressible in TNT! In essence, with symbols in TNT, you can express statements about TNT by applying Godel numbering to itself!!! These ideas are a very distilled summary of Part 1 (GEB), the first 9 chapters of the book.

In Part 2 (EGB), Hofstadter shifts gears and focuses more closely on his ideas of consciousness and intelligence, as well as a more detailed proof of Godel's theorems. We start by looking at different levels of description in computer systems and in brains. Computers are divided into several different levels from bits to machine language to procedural to assembly, with each successive higher level making communication more natural. This raises the interesting question of whether or not the brain is built in a similar set of levels, since this question is deeply tied into the possibility of equivalent intelligence in computers. If computers are supported on the lowest level by hardware, by bits, how is thought supported by the brain, and what is the hardware of the mind? On the lowest level we have neurons, on the highest level we have thought, and Hofstadter suggests that in the middle layers lie symbols, complicated references that trigger and are triggered by other symbols.

We come back to Godel's Incompleteness Theorem when Hofstadter introduces Bloop, FlooP, and GlooP, three computer languages with increasing computation power. FlooP is a Turing Complete language (for those familiar with CS Theory), and GlooP is purely hypothetical (since everything computable is computable by a Turing Machine according to the Church Turing Thesis). An interesting sidenote: some users on wikiwikiweb suggested that adding pimc, pifl, pire (purely theoretical) to FlooP would make GlooP.

From FlooP and GlooP, Hofstadter does a detailed second passthrough Godel's theorem, introducing the Godel string G (the equivalent of the Liar Paradox for TNT), and shows how it is constructed and its undecidability in TNT.

We now have a proof of TNT's incompleteness (assuming consistency of TNT), so the next natural step is to show how this is applicable to ALL formal systems of sufficient power, not just TNT. Using something analogous to Cantor's diagonal method, Hofstadter shows how there is an "essential incompleteness" in all of these systems, and somehow as soon as a system has the basic conditions needed to be consistent or sufficiently powerful Godel's method can be applied.

What are some other examples through which Godel's theorem can be woven? Well, it turns out that there is an interesting analogy between self reference and self replication. Hofstadter compares self reference in Godel's theorem versus self replication in biology , particularly DNA and its self replication, relating sufficiently strong cellular support systems that permit self replication to sufficiently powerful arithmetical formal systems to permit self reference.

In both of these examples, it is the LEVEL mixing that is very important, the presence of self reference that results in the applicability of the Godelian method.

If all computer languages are equal in power to FlooP, and we are trying to simulate mental activity with computing power, how is computing power related to human thought, and can we simulate thought in machinery? If we view thinking as high level descriptions of a system governed by simple and strict rules on a lower level, can we replicate these high level descriptions on a lower level that doesn't consist of neurons? Hofstadter suggests because intelligence doesn't lie in the lower levels, but in higher levels of mixing, mental processes of any sort can be simulated by a computer program written in a Turing Complete language. He calls this the AI version of the Church-Turing Thesis since this belief is at the underpinning of all efforts in AI.

Hofstadter devotes the next two chapters of the book to retrospects and prospects of AI. Interesting retrospects include checkers, chess, translation, music composition, and language programs. I would be really interested in what Hofstadter has to say about current advances in AI, namely self driving cars, Alpha Go, and Google Translate, and how "intelligent" our artificial intelligence is nowadays. Interesting prospects include the AI idea of frames, similar to schemas or layers of context, which leads to a discussion on how different concepts interact and how symbols are created, manipulated, and compared. A question I had was: if we create a beautiful work of art or a piece of music from a computer program, and no listener can distinguish the piece as computer generated rather than human generated, can we still say that the work is not really music or art because the creator (computer) might not necessarily understand it?

The last chapter of the book, Strange Loops, or Tangled Hierarchies, is a wonderful summary of some of the core concepts of the book, looking at other examples of strange loops and how Godel, Escher, and Bach tie together in an Eternal Golden Braid.

A major theme explored in the book and repeated in chapter 20 is

Underneath every tangled hierarchy is an inviolate level, whether we see it or not

Another big point I think I understood better from the second reading is that consciousness, intelligence, sense of self, and all the things that make us confidently divide humans and machines are high level phenomenons. These epiphenomenons are based on interactions between levels, where the top level reaches down towards the bottom level and influences it, while at the same time determined by the bottom level. There is no specific place in the brain where we can point to and say "here is where consciousness is stored! Here is where the Philly cheesesteak symbol is stored!" On a low level, the hardware of our minds is strict, but on a high level, a fuzzy, informal level, epiphenomenon emerge. Therefore, to simulate intelligence in machines, Hofstadter proposes we must be able to simulate this same kind of fuzzy level mixing, this symbol manipulation that we do naturally in our minds in our programs.

This is the eponymous Eternal Golden Braid, and in my mind, this is what Godel Escher Bach is really about. The mathematician, the artist, and the musician are featured precisely because the strands of Godel's Incompleteness Theorem, Escher's art, and Bach's fugues share the integral core of level mixing and self reference, woven together in this book into an Eternal Golden Braid.

So that's my summary of GEB: EGB. There are many ways to view the book; I presented one that's roughly chunked along chapters. I think another interesting (but probably less fruitful) way to chunk the book is by fields. In his book, Hofstadter discusses formal systems, computer architecture, biology, neuroscience, computer languages, AI, mathematics, art, and music, and I probably missed a few more. Perhaps one of the reasons why the core of this book is so hard to grasp is because he touches on so many topics and addresses so many things in a variety of forms.

Like Steve Yegge said, the book is a bit like really good chocolate cake- really rich and yummy, but also really hard to eat more than 1 or 2 servings of. Some people suggest that the reader skip around, read a few pages here and there, and pick up whatever section is interesting to them. I personally think that is a bad strategy for GEB. The author has a goal, a roadmap of ideas connected in a way that was only made clear to me when I read the chapters sequentially and understood each chapter relative to each other.

That being said, I think there are many levels of enjoying the book. The dialogues are a delight to read; his examples are interesting; the connections are brilliant, and I think there's a lot to take away even if not read completely or read out of order.

This is probably one of the most difficult blog posts I've written, and maybe the one with least value, because this book really just defies description. You really just have to read it, or even just parts of it. I hope you find it as interesting and illuminating as I did.